Massachusetts Lottery Arbitrage

“State Treasurer Steven Grossman severely restricted yesterday the number of Cash WinFall lottery tickets any store can sell in a day, closing a loophole that has allowed a handful of high-stakes gamblers to win most of the prizes.

Just three gambling companies collected 1,105 of the 1,605 Cash WinFall prizes statewide after a May drawing, each following a strategy that involved buying hundreds of thousands of dollars worth of the $2 tickets at selected stores over a few days.

Under the new rules, no store will be allowed to sell more than $5,000 worth of Cash WinFall tickets in a single day, making it much harder for the gamblers to continue their high-volume purchases.”

I love the market efficiency here: positive expected value payouts result in the formation of “gambling companies.” The lottery isn’t positive expected value all of the time, though, only during special periods:

“Those times, called “rolldown weeks,’’ take place when the Cash WinFall jackpot grows to roughly $2 million and no ticket wins the jackpot by matching six randomly chosen numbers. The jackpot money is then distributed among the secondary prize-winning tickets, increasing the payoff. For instance, the payoff for matching five numbers rises from $4,000 to a range from $17,500 to as much as $134,767, depending on how many winning tickets are sold.

Several groups – two of them led by highly trained computer scientists from MIT and Northeastern University – formed gambling companies and began pouring hundreds of thousands of dollars into Cash WinFall, a phenomenon lottery officials first noticed in 2005.

The top five groups and individuals playing Cash WinFall collectively win back the cost of their tickets plus $1 million to $6 million in profits each year during rolldowns, without ever winning the jackpot, according to Mohan Srivastava, a Canadian statistician who found a flaw in a Canadian instant game that allowed him to detect winning tickets without scratching them.”

And of course, we have to “blame” these intelligent syndicates who found out how to beat the system:

“News of the quirk in Cash WinFall brought immediate accusations that the game gave an unfair advantage to well-financed players. Overall, the lottery makes a profit from the game, but lottery officials admitted the big payouts made during rolldown weeks are, in effect, subsidized by people who bet at other times when the payoff for winning is far less favorable.”

There is no “unfair advantage” – every ticket is +EV during these periods. Being smart enough to not play the lottery during “normal” periods when ticket purchases have negative EV is not an unfair advantage!* Every lottery that rolls up jackpots each week is like this: the guy who wins the $ 100mm is subsidized by all the others who lost while buying tickets for the “less favorable” $ 10mm, $ 30mm, $ 65mm and $ 85mm un-won jackpots before that.” The difference is that even at $ 100mm, most other lotteries are still not positive EV.

Mrs. Dynamite points out that the State isn’t losing money here: they are simply accelerating the payout of accumulated un-won jackpots, and they probably actually want the gambling syndicates to win quickly so that the State can collect taxes and repeat the cycle. Churn is good for the State – and thus the State seems to have been happy to let this go on for many years: the article notes that they noticed the phenomenon back in 2005! However, when people mis-interpret what’s actually happening and conclude that the lottery is “rigged” in favor of the “well-financed” gambling syndicates, then the State steps in and modifies the rules.

Let’s be clear about one thing: If you buy more lottery tickets, you have a higher chance of winning each prize. You also pay for this “privilege” in the form of higher sunk costs (to buy the tickets). Since I understand mathematics, I do not consider this to be an “advantage,” even if we’re talking about positive expected value trials. Every ticket has the same expected value. It’s a simple mathematical fact that well-financed players can “capture” more expected value (via more tickets). When you buy lots of lottery tickets with slight negative EV, you increase your likelihood of locking in a loss – and when you buy lots of lottery tickets with slight positive EV, you increase your likelihood of locking in a gain. More trials = lower variance from your expected value in your distribution of outcomes.** That said, since lotteries can already easily be argued to be a destructive, regressive “tax” on the poor (the most likely -EV players), I can understand the State’s desire to reform the rules.

-KD

* The article notes that some stores may have broken lottery rules, including giving the Syndicates unfettered access to the lottery machines and letting the Syndicate members run the machines themselves. It should come as no surprise that I do not consider this “fair play.” However, as long as everyone has the opportunity to buy as many tickets as he wants to buy (and State Treasurer Grossman said in the article that this was the case) then I don’t think one can make it a “fairness” issue.

** I am simplifying here, of course, as these lottery ticket “trials” for a single drawing are not independent trials

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